In the Caitanya-caritāmṛta, Ādi-līlā, Chapter Four, this subject matter is explained very clearly. Śrīla Viśvanātha Cakravartī Ṭhākura also notes that no one can describe Kṛṣṇa in fullness. Whatever descriptions we find in Śrīmad-Bhāgavatam are partial explanations of Kṛṣṇa. In conclusion, therefore, the word aṁśena indicates that Lord Viṣṇu is a partial representation of Kṛṣṇa, not that Kṛṣṇa is a partial representation of Viṣṇu.
Śrīla Sanātana Gosvāmī's Vaiṣṇava-toṣaṇī has explained the word dharma-śīlasya. The exact meaning of dharma-śīla is "an unadulterated devotee." Real dharma consists of full surrender to Kṛṣṇa (sarva-dharmān parityajya mām ekaṁ śaraṇaṁ vraja (BG 18.66)). One who has fully surrendered to Kṛṣṇa is actually religious. One such religious person was Mahārāja Parīkṣit. Anyone who accepts the principle of surrender to the lotus feet of the Lord, giving up all other systems of religion, is actually dharma-śīla, perfectly religious.
The word nivṛtta-tarṣaiḥ refers to one who no longer has any material desires (sarvopādhi-vinirmuktam (CC Madhya 19.170)). One may have many material desires because of contamination in this material world, but when one is completely free from all material desires, he is called nivṛtta-tṛṣṇa, which indicates that he no longer has any thirst for material enjoyment.